Given $ \overrightarrow{OA}\perp\overrightarrow{OC}$, $ m \angle AOB = 6x - 71$, and $ m \angle BOC = 5x - 4$, find $m\angle BOC$. $O$ $A$ $C$ $B$
Answer: From the diagram, we see that together ${\angle AOB}$ and ${\angle BOC}$ form ${\angle AOC}$ , so $ {m\angle AOB} + {m\angle BOC} = {m\angle AOC}$ Since we are given that $\overrightarrow{OA}\perp\overrightarrow{OC}$ , we know ${m\angle AOC = 90}$ Substitute in the expressions that were given for each measure: $ {6x - 71} + {5x - 4} = {90}$ Combine like terms: $ 11x - 75 = 90$ Add $75$ to both sides: $ 11x = 165$ Divide both sides by $11$ to find $x$ $ x = 15$ Substitute $15$ for $x$ in the expression that was given for $m\angle BOC$ $ m\angle BOC = 5({15}) - 4$ Simplify: $ {m\angle BOC = 75 - 4}$ So ${m\angle BOC = 71}$.